202 research outputs found
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity
This paper deals with the existence and the asymptotic behavior of
non-negative solutions for a class of stationary Kirchhoff problems driven by a
fractional integro-differential operator and involving a
critical nonlinearity. The main feature, as well as the main difficulty, of the
analysis is the fact that the Kirchhoff function can be zero at zero, that
is the problem is degenerate. The adopted techniques are variational and the
main theorems extend in several directions previous results recently appeared
in the literature
On the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity
We consider the fractional Schr\"{o}dinger-Kirchhoff equations with
electromagnetic fields and critical nonlinearity
where as and
is the fractional magnetic operator with , is a continuous nondecreasing
function, and are the electric and the magnetic potential,
respectively. By using the fractional version of the concentration compactness
principle and variational methods, we show that the above problem: (i) has at
least one solution provided that ; and (ii) for any
, has pairs of solutions if , where and are
sufficiently small positive numbers. Moreover, these solutions as
Infinitely many periodic solutions for a class of fractional Kirchhoff problems
We prove the existence of infinitely many nontrivial weak periodic solutions
for a class of fractional Kirchhoff problems driven by a relativistic
Schr\"odinger operator with periodic boundary conditions and involving
different types of nonlinearities
Concentrating solutions for a fractional Kirchhoff equation with critical growth
In this paper we consider the following class of fractional Kirchhoff
equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll}
\left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u
\quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0
&\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where
is a small parameter, are constants, , is the fractional critical
exponent, is the fractional Laplacian operator, is a
positive continuous potential and is a superlinear continuous function with
subcritical growth. Using penalization techniques and variational methods, we
prove the existence of a family of positive solutions which
concentrates around a local minimum of as .Comment: arXiv admin note: text overlap with arXiv:1810.0456
A fractional Kirchhoff problem involving a singular term and a critical nonlinearity
In this paper we consider the following critical nonlocal problem
\left\{\begin{array}{ll}
M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s
u = \displaystyle\frac{\lambda}{u^\gamma}+u^{2^*_s-1}&\quad\mbox{in } \Omega,\\
u>0&\quad\mbox{in } \Omega,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminus\Omega,
\end{array}\right. where is an open bounded subset of
with continuous boundary, dimension with parameter ,
is the fractional critical Sobolev exponent, is a
real parameter, exponent , models a Kirchhoff type
coefficient, while is the fractional Laplace operator. In
particular, we cover the delicate degenerate case, that is when the Kirchhoff
function is zero at zero. By combining variational methods with an
appropriate truncation argument, we provide the existence of two solutions
- …